The circulation of the field F = -2xi - 2yj around the closed semicircular path can be calculated using Stoke's theorem. Since the path consists of a semicircular arch and a line segment, we can divide the path into two parts and calculate the circulation for each part separately.
For the semicircular arch, we can use the equation of the circle to parameterize the path:
r(t) = a(cos(t)i + sin(t)j), 0 <= t <= pi
where a is the radius of the circle. The unit tangent vector along the path is:
T(t) = (-sin(t)i + cos(t)j)
The circulation along the semicircular arch is:
C1 = int_C1 F . dr = int_0^pi (-2a cos(t)i - 2a sin(t)j) . (-a sin(t)i + a cos(t)j) dt
= int_0^pi 2a^2 dt = 2a^2 pi
For the line segment, we can use the endpoints of the path to parameterize the path:
r(t) = (1-t)(a)i + tj, 0 <= t <= 1
The unit tangent vector along the path is:
T(t) = -a i + j
The circulation along the line segment is:
C2 = int_C2 F . dr = int_0^1 (-2a i - 2t j) . (-a i + j) dt
= int_0^1 (2at - 2t) dt = a - 1
The total circulation along the closed semicircular path is:
C = C1 + C2 = 2a^2 pi + a - 1
The flux of the field F = -2xi - 2yj across the closed semicircular path can be calculated using the divergence theorem. Since the path encloses a region in the xy-plane, we can use the 2D form of the divergence theorem:
int_S F . n dA = int_V div(F) dV
where S is the boundary of the region enclosed by the path, n is the unit outward normal vector to S, and V is the region enclosed by S.
The divergence of F is:
div(F) = -2
Since F is a constant vector field, the flux across the closed semicircular path is:
Phi = int_S F . n dA = int_V